I originally meant for this blog to be an outlet of some topics in math that happened to be a bit too abstract for the general audience. After all, it would have made little sense to post such topics on my main blog, which has a more diverse audience in mind.
This blog went pretty low on my priority list, especially as I got busier this year, but hey, we’re all busy, right? I might be posting sporadically on this blog in 2014, but the main action will be at the main blog (again). Happy Holidays!
The following example was brought up in two different classes that I am taking, within a couple days of each other. It is the classic example of an immeasurable set on the interval [0,1]. Now in math, words like measure and measurable have technical definitions, and lead to bizarre results like the Banach-Tarski paradox. I’m going to give a fairly informal explanation here. Continue reading
Wrong answer: To get to the other side.
Right answer: It couldn’t, because there is no other side.
Mathematician’s answer: It couldn’t, because chickens cannot cross non-orientable manifolds.
Thanks to Aaron L. for this one.
Everyone logician knows, “The first rule of Tautology Club is the first rule of Tautology Club.”
Some friends and I made some rules for Paradox Club:
- The first rule of Paradox Club is not the first rule of Paradox Club.
- The second rule of Paradox Club is, there is no second rule of Paradox Club.
- The third rule of Paradox Club is, the fourth rule of Paradox Club must sometimes be followed.
- The fourth rule of Paradox Club is, the third rule of Paradox Club must never be followed.
- “Yields falsehood when preceded by its quotation when listed as the fifth rule of Paradox Club,” yields falsehood when preceded by its quotation when listed as the fifth rule of Paradox Club.
- The sixth rule of Paradox Club cannot be proven to be the sixth rule of Paradox Club.
- The seventh rule of Paradox Club invalidates precisely the rules of Paradox Club which do not invalidate themselves.
- The eighth rule of Paradox Club is, do not follow any rules of Paradox Club.
Here are a number of misleading sequences. I have given the first few numbers of each sequence, and your mission, if you choose to accept it, is to guess the next number.
1. A Doubling Dilemma
1, 2, 4, 8, 16, ___?
Did you get 32 from doubling?
The answer is actually 31. Continue reading
A link to this article in pdf format: [pdf]
Here is a difficult probability question:
Suppose you are standing on an infinitely large square grid at the point (0,0), and suppose that you can see infinitely far but cannot see through grid points. Given a random grid point z = (x,y), where x and y are integers, what is the chance you can see z?
Here is the link to the paper [pdf].
Abstract: The main statistical result of this paper is that of the reddit users who have upvoted at least one post, 31.4% of them have upvoted only one post and, moreover, have been the only person to upvote it. Conversely, 22.6% of posts which have been voted on at all have received only downvotes and no upvotes. In addition, 67.4% of users are connected in a giant component of upvoting common posts, while 91.4% of posts are connected by having been upvoted by common users.